CN110744552A - Flexible mechanical arm motion control method based on singular perturbation theory - Google Patents

Flexible mechanical arm motion control method based on singular perturbation theory Download PDF

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CN110744552A
CN110744552A CN201911178147.9A CN201911178147A CN110744552A CN 110744552 A CN110744552 A CN 110744552A CN 201911178147 A CN201911178147 A CN 201911178147A CN 110744552 A CN110744552 A CN 110744552A
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mechanical arm
controller
flexible mechanical
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CN110744552B (en
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许一鸣
豆康康
王丽
方建勇
张亚
王凯
陈伟
惠小霞
费琪
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716th Research Institute of CSIC
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator

Abstract

The invention belongs to the field of flexible mechanical arms, and particularly relates to a flexible mechanical arm motion control method based on a singular perturbation theory. The method specifically comprises a combined control method combining sliding mode variable structure control and optimal control. Firstly, considering the influence of external disturbance on a system, designing a sliding mode controller to realize the track tracking of the flexible mechanical arm; meanwhile, the vibration quantity of the system under the slow time scale is estimated by adopting a least square method, so that a foundation is laid for the design of a controller of the fast time scale system; then, under a fast time scale, based on an optimal control theory, designing a vibration controller to realize vibration suppression of the flexible mechanical arm; and finally, combining the flexible mechanical arm and the flexible mechanical arm by utilizing a singular perturbation principle to realize the combined motion control of the flexible mechanical arm and the flexible mechanical arm. The invention provides a novel flexible mechanical arm combination control method based on sliding mode control and optimal control, and the method can be used for realizing double control targets of position tracking and vibration suppression of a flexible mechanical arm.

Description

Flexible mechanical arm motion control method based on singular perturbation theory
Technical Field
The invention belongs to the field of flexible mechanical arms, and particularly relates to a flexible mechanical arm motion control method based on a singular perturbation theory.
Background
The robot technology has urgent needs in the fields of aerospace, ocean exploration, industrial manufacturing, life entertainment and the like, and is one of key research contents in the field of industrial manufacturing. The deep fusion of the artificial intelligence technology and the advanced control method is an effective means for improving the stability, the self-adaptability, the anti-interference performance and the optimality of a robot system in a complex working environment and realizing the motion control with high precision and high reliability, and has attracted wide attention in the industry and academia.
The mechanical arm is an important branch of the robot field and is always a research hotspot of experts and scholars. The mechanical arm simulates the arm work of a person according to the instruction, and completes the mechanical grabbing, carrying and other works. The traditional mechanical arm mainly improves the positioning accuracy by increasing the rigidity of materials, but the larger rigidity can cause the problems of lagging working positioning, large energy consumption, low speed, high cost and the like.
The flexibility of the connecting rod is increased, the problems can be effectively solved, and the contradiction between high speed and accuracy in operation is solved. At present, a mechanical arm which has joint flexibility or arm lever flexibility and can meet the requirements of light weight and high speed is generally called as a flexible mechanical arm. The flexible robot arm has the advantages of light weight, high speed, low energy consumption and the like, and has an important position in the field of robots, particularly in the field of space robots, so that the flexible robot arm is concerned and researched by extensive experts and scholars. The flexible mechanical arm has the characteristics of high coupling, infinite dimension, continuous distribution parameters and the like, and has larger deformation in the motion process, so that the motion rule of the flexible mechanical arm is essentially different from that of the traditional rigid mechanical arm. Therefore, the flexible mechanical arm is more complex than the traditional rigid mechanical arm in the aspects of dynamic modeling and motion control design. Therefore, how to establish an effective mathematical model and how to design a high-performance controller to realize motion control of the flexible mechanical arm is two important research directions in the field of flexible mechanical arms.
Due to the flexible existence of the flexible mechanical arm structure, the system has the characteristics of rigid-flexible coupling, nonlinearity, uncertain parameters, infinite dimension and the like, and the characteristics bring great difficulty to the dynamic modeling and control of the flexible mechanical arm. In order to characterize the above characteristics, a partial differential equation of the flexible mechanical arm is generally established by a Lagrange method, a Newton-Euler method, a Kane equation method and the like, although the model is accurate, the equation is not easy to solve, and it is difficult to directly design a controller on the basis of the partial differential equation to realize motion control of the flexible mechanical arm. On the basis of the partial differential equation, a large number of scholars describe the flexible deformation of the flexible mechanical arm by using an assumed modal method, a finite element method, a concentrated mass method and the like, further convert a partial differential dynamic equation of the flexible mechanical arm into a general ordinary differential equation, and design a controller on the basis.
The flexible mechanical arm is a rigid-flexible coupling, infinite dimension and parameter uncertainty system. Due to the existence of load quality and external interference, the flexible mechanical arm is easy to bend and stretch and deform in the moving process, so that the vibration problem of the flexible mechanical arm is caused. The existence of flexible vibration greatly reduces the positioning precision of the mechanical arm.
Disclosure of Invention
The invention aims to provide a flexible mechanical arm combination control method based on a singular perturbation theory, which solves the problem of motion control of a flexible mechanical arm under multiple time scales.
The technical solution for realizing the purpose of the invention is as follows: a flexible mechanical arm motion control method based on a singular perturbation theory comprises the following steps:
step (1): decomposing the flexible mechanical arm system into subsystems under different time scales by using a singular perturbation theory, and designing a controller on the basis of the subsystems under different time scales;
step (2): considering the influence of external disturbance on a system, designing a sliding mode controller, namely a slow controller, based on a sliding mode variable structure under a slow time scale, and realizing the track tracking of the flexible mechanical arm;
and (3): the vibration quantity of the system under the slow time scale is estimated by adopting a least square method, so that a foundation is laid for the design of a controller of the fast time scale system;
and (4): under a fast time scale, a vibration controller, namely a fast controller is designed based on an optimal control theory, so that vibration suppression of the flexible mechanical arm is realized;
and (5): and (3) combining the sliding mode controller in the step (2) and the vibration controller in the step (4) by using a singular perturbation principle to realize the combined motion control of the flexible mechanical arm.
The flexible mechanical arm is decomposed into a slow subsystem for describing rigid motion and a fast subsystem for describing flexible vibration based on the singular perturbation theory, the controllers are respectively designed under different time scales, the flexible mechanical arm trajectory tracking control is realized, the flexible vibration is inhibited, the method is simple to realize, and the control precision is high. Under a slow time scale, the motion process of the flexible mechanical arm is easily influenced by external slow time-varying interference, and system parameters are uncertain and fixedly existing, so that the tracking precision is reduced. The sliding mode control enables a system to be converged along the designed hyperplane direction under a variable structure control strategy by designing a switching plane, and has good robustness on parameter uncertainty and external disturbance meeting matching conditions. The sliding mode variable structure design controller is used for realizing high robustness and anti-interference performance on track tracking of the flexible mechanical arm. Under the fast time scale, an optimal controller is designed to realize vibration suppression of the flexible mechanical arm, and experimental results show that the method has a good control effect, but the method does not consider the influence of a slow vibration state on a system, so that the control precision is reduced. Therefore, the controller is designed respectively under different time scales based on the singular perturbation theory, the influence of external interference and parameter change on the system is fully considered, and the influence of a slow vibration state on the system is fully considered under a fast time scale.
Compared with the prior art, the invention has the remarkable advantages that:
(1) a novel flexible mechanical arm combination control method based on sliding mode control and optimal control is provided, and experimental results show that the method has a better control effect, and enriches and expands the motion control theory of the flexible mechanical arm;
(2) the slow vibration state is estimated by using a least square method, a fast dynamic design controller is reconstructed, and the control precision is higher;
(3) the singularity perturbation theory is used for proving that the closed loop of the system is stable, and the method has a good control effect and high control precision according to an experimental result.
Drawings
Fig. 1 is a flowchart of a method for controlling the movement of a flexible mechanical arm according to the present invention.
FIG. 2 is a diagram of a flexible robot arm model according to the present invention.
FIG. 3 is a saturation function diagram of the present invention.
Fig. 4 is a flow chart of the simulation of the motion control of the flexible mechanical arm according to the present invention.
Fig. 5 is a track following diagram of the flexible mechanical arm of the invention.
FIG. 6 is a first order modal response graph of the present invention.
FIG. 7 is a second order modal response graph of the present invention.
Detailed Description
The following describes a motion control method of a flexible robot arm according to an embodiment of the present invention with reference to the drawings.
As shown in fig. 1, a method for controlling the motion of a flexible mechanical arm based on the singular perturbation theory in an embodiment of the present invention includes the following steps:
s1: the method comprises the following steps of establishing a flexible mechanical arm dynamic model based on a Lagrange method and an assumed modal method, decomposing the dynamic model into subsystems under different time scales by using a singular perturbation theory, and specifically comprising the following steps:
the method comprises the following steps: establishing a flexible mechanical arm dynamic model
Figure BDA0002290521820000031
Wherein u is system input, theta is mechanical arm rotation angle, q is system vibration mode, M is positive definite matrix, and G isθ、GqIs a non-linear term, K is a stiffness matrix, d is an external disturbance, and d<D, D is the interference upper bound. Specifically, the deployment is
Figure BDA0002290521820000041
Wherein the content of the first and second substances,
Figure BDA0002290521820000042
Figure BDA0002290521820000043
Figure BDA0002290521820000044
wherein, JhIs motor torque, L is mechanical arm length, m1，m2The arm and the tail end load mass respectively, E is the elastic modulus, and I is the section moment of inertia.
And step two, because the flexible vibration of the flexible mechanical arm is changed relatively fast to the rigid rotation, the flexible mechanical arm system has the characteristic of double time scales, wherein epsilon is defined as 1/K, epsilon z is defined as q, β is defined as epsilon K,
Figure BDA0002290521820000045
k is the minimum value of elements in the rigidity matrix K, epsilon is a singular perturbation parameter, β and z are new definition variables, and the rigidity matrix K is obtained
Figure BDA0002290521820000046
Based on a singular perturbation theory, let epsilon be 0, and obtain the system state under a slow time scale:
Figure BDA0002290521820000047
the external disturbance is considered to change slowly, so that the slowly changing disturbance exists only in the slow subsystem. Further solving a slow subsystem model:
Figure BDA0002290521820000048
where the superscript s denotes slow dynamics, usIs the output of the controller of the slow subsystem,is an estimate of theta on a slow time scale. Further collated and combined to define
Figure BDA00022905218200000410
The above formula can be rewritten as:
Figure BDA0002290521820000051
step three: introducing multiple time scale variables
Figure BDA0002290521820000052
At the τ time scale, the system state at the slow time scale can be considered constant, i.e.:
Figure BDA0002290521820000053
dzs/dτ＝d2zs/dτ2＝0
defining a variable z taking into account the dual time scale characteristics of the flexible manipulatorf＝z-zsIn the boundary layer area, let epsilon equal to 0, the fast subsystem model of the flexible manipulator can be obtained:
Figure BDA0002290521820000054
ufand outputting the data to the controller of the fast subsystem. According to the singular perturbation theory, the relationship between the slow subsystem and the fast subsystem and the original system is as follows:
q＝1/k(zs+zf)+O(ε)
where O (ε) is the high order of ε, infinitesimally small. As can be seen from the formula, the state of the flexible mechanical arm slow subsystem
Figure BDA0002290521820000056
The vibration mode of the original system is approximated to the state z of the slow subsystemsAnd fast subsystem state zfAnd (4) summing.
S2: under a slow time scale, designing a controller (a slow controller) based on a sliding mode variable structure control theory, and specifically comprising the following steps of:
the method comprises the following steps: the method considers the existence of external interference, designs the slow controller based on the sliding mode control algorithm under the slow time scale, and inhibits the sliding mode buffeting by designing the quasi-sliding mode.
For the slow subsystem describing rigid body motion, the trajectory tracking error is defined:
Figure BDA0002290521820000057
Figure BDA0002290521820000058
where e (t) is the tracking error, θdIs the desired angle signal.
Step two: designing sliding mode function
Figure BDA0002290521820000059
Wherein the error convergence rate is dependent on the undetermined parameter c, c>0. Designing a sliding mode controller:wherein sgn(s) is a sign function, η is a undetermined parameter, and D is an interference upper bound.
Step three: defining a Lyapunov function
Figure BDA0002290521820000062
Then
Figure BDA0002290521820000063
Obtaining:
Figure BDA0002290521820000064
according to the LaSalle invariance principle, the closed-loop system asymptotically stabilizes, i.e., when t → ∞, s → 0, the convergence rate depends on the parameter η.
Step four: due to the discontinuous switching characteristic of the sliding mode control, the system is easy to generate buffeting. In order to eliminate the influence of sliding mode buffeting on a system, a quasi-sliding mode is designed to be an effective control strategy. The saturation function sat(s) is used for replacing the sign function sgn(s), switching control is adopted outside the boundary layer, linear feedback control is adopted in the boundary layer, the motion track is limited in a certain neighborhood of an ideal sliding mode, and therefore buffeting can be effectively avoided. The saturation function equation sat(s) is designed as:wherein, delta is a boundary layer, and the controller is improved as follows:
s3: under the slow time scale, the flexible mechanical arm vibration state is estimated based on the least square method, and the method specifically comprises the following steps:
the method comprises the following steps: the least square method is one of the important research methods in the field of system identification, and system parameter values are obtained by continuously using new data to update. The input and output of the identification model are as follows: y (i) ═ θ1x1(i)+θ2x2(i)+θ3x3(i)+...+θnxn(i) 1,2,3.. m, wherein xn(i) For the nth input sampled by the system at time i, y (i) for the system model sampled output at time i, θnFor the nth parameter to be identified, writing the above formula as a matrix form: y ═ X θ. Wherein the content of the first and second substances,
Figure BDA0002290521820000071
m is the number of sampling input and output sequences, and n is the number of parameters to be identified.
Step two: according to the least square law, let the theta estimate be
Figure BDA0002290521820000072
Defining an error: if γ is the smallest, θ identified at this time is the best fitting parameter. The fitting parameters closest to the true values are identified by minimizing the sum of the squares of the differences, i.e.:to minimize the value of the above equation, let
Figure BDA0002290521820000074
According to the quadratic minimum theorem, the derivation is carried out on the formula:
Figure BDA0002290521820000075
obtaining:
Figure BDA0002290521820000076
the minimum value is required to be sufficient according to the quadratic form because
Figure BDA0002290521820000077
The above formula satisfies the positive definite condition.
Step three: according to the model, the vibration state of the flexible mechanical arm comprises zsAnd zfTwo parts, to design a fast controller must first obtain zs. On a slow time scale, the invention chooses to estimate z using a least squares methods. The flexible mechanical arm vibration amount can be described as follows under a slow time scale: z is a radical ofs＝a+busAnd a and b are parameters to be estimated.
Considering the random error that may exist, the above equation can be rewritten as
Figure BDA0002290521820000078
Wherein v isiIn order to be a random error,
Figure BDA0002290521820000079
the method is used for measuring the vibration data of the flexible mechanical arm under the slow time scale.Selecting a performance index function for the input of the ith system iteration system according to a least square method:
Figure BDA00022905218200000711
to minimize the performance indicator function, let:
Figure BDA00022905218200000712
obtaining approximate values of the parameters a and b:
further obtaining the vibration z of the flexible mechanical arm under the slow time scales：
Figure BDA0002290521820000082
Step four: according to the singular perturbation theory, reconstructing a fast vibration state: z is a radical off＝k·q-zs。
S4, designing a controller (fast controller) based on the optimal control theory under the fast time scale, which specifically comprises the following steps:
the method comprises the following steps: the fast time scale model for describing the vibration characteristics of the system has a state space expression as follows:
Figure BDA0002290521820000083
wherein
Figure BDA0002290521820000084
Step two: designing an optimal controller: u. off-Kx, so that the following performance indicator function is minimized
Figure BDA0002290521820000085
Wherein Q is QT>0,R＝RT>0，(A,Q1/2) It is considerable.
Step three: according to the optimal control theory, the following are: u. off＝-Kx＝-R-1BPx, where K is the optimal feedback controller gain and P is the algebraic Riccati equation ATP+PA-PBR-1BTP + Q ═ 0 positive solution.
Step four: defining the Lyapunov function: x isTPx, solving an algebraic Riccati equation, and differentiating the above equation to obtain:since R, P is a symmetric positive definite matrix and Q is a positive definite matrix, the method obtains
Figure BDA0002290521820000087
Therefore, under the action of the designed fast controller, the fast subsystem is gradually stabilized.
S5: based on singular perturbation theory, design a flexible mechanical arm motion controller based on singular perturbation theory specifically includes:
combining a slow controller based on sliding mode control designed under a slow time scale with a fast controller designed under a fast time scale to obtain: u (t) ═ us(t)+ufAnd (t) realizing the motion control of the flexible mechanical arm system. In order to verify the effectiveness of the method, the effectiveness of the method is verified in a simulation mode under a Matlab environment. The method utilizes a singular perturbation theory to obtain a rigid-flexible coupling decomposition model of the flexible mechanical arm. Under a slow time scale, the rotation angle of the flexible mechanical arm is approximate to a rigid motion angle. Considering the influence of external disturbance on a system, a quasi-sliding mode-based sliding mode control method is designed to realize track tracking control on the flexible mechanical arm. Under the fast time scale, a fast vibration state is reconstructed by using a least square method, and an optimal controller is designed to realize vibration suppression of the flexible mechanical arm. Selecting an ideal tracking trajectory thetadThe ideal vibration state is 0. Firstly, for the slow subsystem (3-8), the sliding mode control shown in the formula (3-19) is designedThe controller design parameter is selected as c 12, η 0.5, the saturation function critical parameter is delta 0.2, and the slow vibration state is obtained by the least square method
Figure BDA0002290521820000091
Under the fast time scale, the controller is designed by utilizing the optimal control theory to realize vibration suppression, and a corresponding matrix of a performance index function is selected: q ═ diag (1,0.1,1,0.1), R ═ I. According to the linear quadratic optimal control theory, the optimal feedback gain matrix K [ -0.1084-0.65780.12930.1856 ] is obtained]。
Fig. 4 is a flow chart of the flexible mechanical arm motion control simulation. Fig. 5 shows the actual motion trajectory of the mechanical arm under the action of the combined controller, and it can be seen from the figure that the trajectory tracking control effect is better under the action of the controller designed by the invention. Fig. 6 and 7 are first two-order modal vibration suppression curves of the flexible manipulator, and it can be seen from the graphs that the vibration suppression effect of the flexible manipulator is better under the action of the optimal controller designed by the invention.

Claims (6)

1. A flexible mechanical arm motion control method based on a singular perturbation theory is characterized by comprising the following steps:
step (1): decomposing the flexible mechanical arm system into subsystems under different time scales by using a singular perturbation theory, and designing a controller on the basis of the subsystems under different time scales;
step (2): considering the influence of external disturbance on a system, designing a sliding mode controller, namely a slow controller, based on a sliding mode variable structure under a slow time scale, and realizing the track tracking of the flexible mechanical arm;
and (3): the vibration quantity of the system under the slow time scale is estimated by adopting a least square method, so that a foundation is laid for the design of a controller of the fast time scale system;
and (4): under a fast time scale, a vibration controller, namely a fast controller is designed based on an optimal control theory, so that vibration suppression of the flexible mechanical arm is realized;
and (5): and (3) combining the sliding mode controller in the step (2) and the vibration controller in the step (4) by using a singular perturbation principle to realize the combined motion control of the flexible mechanical arm.
2. The method according to claim 1, wherein the decomposition of step (1) into subsystems at different time scales comprises in particular the steps of:
step (11): establishing a flexible mechanical arm dynamic model
Figure FDA0002290521810000011
Wherein u is system input, theta is mechanical arm rotation angle, q is system vibration mode, M is positive definite matrix, and G isθ、GqIs a non-linear term, K is a stiffness matrix, d is an external disturbance, and d<D, D is an interference upper bound;
the flexible mechanical arm system has a dual time scale characteristic, and is defined as epsilon-1/K, epsilon-z-q, β -epsilon-K,k is the minimum value of the elements in the stiffness matrix K, epsilon is a singular perturbation parameter, β, z is a new defined variable, and the following are obtained:
Figure FDA0002290521810000014
step (12): based on a singular perturbation theory, let epsilon be 0, and obtain the system state under a slow time scale:
Figure FDA0002290521810000015
and (3) slowly changing disturbance only exists in a slow subsystem, and further solving a slow subsystem model:
Figure FDA0002290521810000021
where the superscript s denotes slow dynamics, usIs the output of the controller of the slow subsystem,
Figure FDA0002290521810000022
is an estimated value of theta under a slow time scale;
step (13): introducing multiple time scale variablesAt the τ time scale, the system state at the slow time scale can be considered constant, i.e.:
Figure FDA0002290521810000024
dzs/dτ＝d2zs/dτ2＝0
defining a variable z taking into account the dual time scale characteristics of the flexible manipulatorf＝z-zsIn the boundary layer area, let epsilon be 0, further obtain the fast subsystem model of the flexible mechanical arm:
Figure FDA0002290521810000025
uffor the output of the fast subsystem controller, according to the singular perturbation theory, the relationship between the slow and fast subsystems and the original system is as follows:
Figure FDA0002290521810000026
q＝1/k(zs+zf)+O(ε)
wherein O (epsilon) is high-order infinitesimal of epsilon, and the state of the flexible mechanical arm slow subsystem can be seen from the formula
Figure FDA0002290521810000027
The vibration mode of the original system is approximated to the state z of the slow subsystemsHekuaizi systemSystem state zfAnd (4) summing.
3. The method according to claim 2, wherein the step (2) of designing the sliding mode controller based on the sliding mode variable structure at the slow time scale specifically comprises the following steps:
step (21): considering the existence of external interference, designing a slow controller based on a sliding mode control algorithm under a slow time scale, and restraining sliding mode buffeting by designing a quasi-sliding mode;
for the slow subsystem describing rigid body motion, the trajectory tracking error is defined:
Figure FDA0002290521810000031
where e (t) is the tracking error, θdIs a desired angle signal;
step (22): designing sliding mode function
Figure FDA0002290521810000032
Wherein the error convergence rate is dependent on the undetermined parameter c, c>0，
Designing a sliding mode controller:
Figure FDA0002290521810000033
wherein sgn(s) is a sign function, η is a undetermined parameter, and D is an interference upper bound;
step (23): defining a Lyapunov function
Figure FDA0002290521810000034
ThenObtaining:
according to the LaSalle invariance principle, the closed-loop system asymptotically stabilizes, i.e., when t → ∞, s → 0, the convergence rate depends on the parameter η;
step (24): due to the discontinuous switching characteristic of sliding mode control, the system is easy to generate buffeting, a saturation function sat(s) is used for replacing a sign function sgn(s), switching control is adopted outside a boundary layer, linear feedback control is adopted in the boundary layer, and the motion track is limited in a certain neighborhood of an ideal sliding mode to avoid buffeting;
the saturation function equation sat(s) is designed as:
Figure FDA0002290521810000037
wherein, delta is a boundary layer, and the controller is improved as follows:
Figure FDA0002290521810000038
4. the method according to claim 3, wherein the step (3) adopts a least square method to estimate the vibration quantity of the system under the slow time scale, and lays a foundation for the design of the controller of the fast time scale system, and specifically comprises the following steps:
step (31): according to the fast and slow subsystem models, the vibration state of the flexible mechanical arm comprises zsAnd zfTwo parts, to design a fast controller must first obtain zsEstimating z using least squares at a slow time scalesThe vibration quantity of the flexible mechanical arm is described as follows under a slow time scale: z is a radical ofs＝a+busWherein a and b are parameters to be estimated;
step (32): considering the random error that may exist, the above equation can be rewritten asWherein v isiIn order to be a random error,the vibration data of the flexible mechanical arm measured under the slow time scale,
Figure FDA0002290521810000043
selecting a performance index function for the input of the ith iteration system according to a least square method:
Figure FDA0002290521810000044
step (33): to minimize the performance indicator function, let:
Figure FDA0002290521810000045
obtaining approximate values of the parameters a and b:further solving the vibration quantity of the flexible mechanical arm under the slow time scale:
Figure FDA0002290521810000047
step (34): according to the singular perturbation theory, reconstructing a fast vibration state: z is a radical off＝k·q-zs。
5. The method according to claim 4, wherein the vibration controller of step (4) is designed based on an optimal control theory, and specifically comprises the following steps;
step (41): the fast time scale model for describing the vibration characteristics of the system has a state space expression as follows:
Figure FDA0002290521810000048
wherein
Figure FDA0002290521810000049
I2×2Is a 2 x 2 unit matrix;
step (42): designing an optimal controller: u. off-Kx, so that the following performance indicator function is minimized
Figure FDA0002290521810000051
Wherein Q is QT>0,R＝RT>0，(A,Q1/2) It is considerable.
Step (43): according to the optimal control theory, the following are: u. off＝-Kx＝-R-1BPx, where K is the optimal feedback controller gain and P is the algebraic Riccati equation ATP+PA-PBR-1BTP + Q ═ 0 positive definite solution;
step (44): defining the Lyapunov function: x isTPx, solving an algebraic Riccati equation, and differentiating the above equation to obtain:
Figure FDA0002290521810000052
since R, P is a symmetric positive definite matrix and Q is a positive definite matrix, the method obtains
Figure FDA0002290521810000053
Therefore, under the action of the designed fast controller, the fast subsystem is gradually stabilized.
6. The method according to claim 5, wherein the step (5) combines the sliding mode controller of the step (2) and the vibration controller of the step (4) by using a singular perturbation principle to realize the combined motion control of the flexible mechanical arm, and specifically comprises the following steps:
combining the slow controller based on sliding mode control designed in the slow time scale of the step (2) with the fast controller designed in the fast time scale of the step (4) to obtain: u (t) ═ us(t)+ufAnd (t) realizing the motion control of the flexible mechanical arm system.
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